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GENUS DISTRIBUTIONS of STAR-LADDERS Newly Developed GENUS DISTRIBUTIONS OF STAR-LADDERS YICHAO CHEN, JONATHAN L. GROSS, AND TOUFIK MANSOUR Abstract. Star-ladder graphs were introduced by Gross in his development of a quadratic-time algorithm for the genus distribution of a cubic outerplanar graph. This paper derives a formula for the genus distribution of star-ladder graphs, using Mohar's overlap matrix and Chebyshev polynomials. Newly developed methods have led to a number of recent papers that de- rive genus distributions and total embedding distributions for various families of graphs. Our focus here is on a family of graphs called star-ladders. 1. Introduction Genus distributions problems have frequently been investigated in the past quarter century, since the topic was inaugurated by Gross and Furst [6]. The contributions include [1,5,8,9, 10, 11, 12, 14, 15, 16, 18, 19, 20, 22, 23, 24, 25, 26] and [27]. Gross [11] presents a quadratic-time algorithm for computing the genus distribution of any cubic outerplanar graph. He analyzes the structure of any cu- bic outerplanar graph and finds that such a graph can be obtained by a series of iterated edge-amalgamations of a new classes of graphs called star-ladders, so as to form a tree of star-ladders. Thus, beyond the direct interest in a closed formula for the genus distribution of star-ladders, such a formula is possibly a step toward a closed formula for the genus distribution of the cubic outerplanar graphs. Our closed formula in this paper for the genus distribution of star-ladders is derived with the aid of Mohar's overlap matrices [17]. 1.1. Star-ladders. An n-rung closed-end ladder Ln can be obtained by tak- ing the graphical cartesian product of an n-vertex path with the complete graph K2, and then doubling both its end edges. The new rungs obtained thereby are called end-rungs. Figure1 presents a 4-rung closed-end ladder. In [5], Furst, Gross, and Statman obtained a closed formula for the genus distribution of closed- end ladders. 2000 Mathematics Subject Classification. Primary: 05C10; Secondary: 30B70, 42C05. Key words and phrases. graph embedding; genus distribution; star-ladders; overlap matrix; Chebyshev polynomials. The work of the first author was partially supported by NNSFC under Grant No. 10901048. 1 2 YICHAO CHEN, JONATHAN L. GROSS, AND TOUFIK MANSOUR Figure 1. The 4-rung closed-end ladder L4. For an k-tuple of non-negative integers U = (n1; n2; : : : ; nk) the star-ladder with signature U is the graph SLn1;n2;:::;nk obtained from the cycle graph C2k, with consecutive edges labeled e1; e2; : : : ; e2k as follows: (1) For each i ≤ k such that ni = 0, double the edge e2i. (2) For each of the ladders Ln1 ;Ln2 ;:::;Lnk such that ni > 0, • subdivide one end rung of Lni into three parts, and take the middle third as the root-edge; • amalgamate Lni across its newly created root edge to the edge e2i. The star-ladder SL2;1;0 is shown in Figure2. Figure 2. The star-ladder SL2;1;0. 1.2. Genus polynomial. It is assumed that the reader is somewhat familiar with the basics of topological graph theory, as found in Gross and Tucker [7]. All graphs considered in this paper are connected. A graph G = (V (G);E(G)) is permitted to have both loops and multiple edges. A surface is a compact 2-manifold without boundary. In topology, surfaces are classified into the orientable surfaces Sg, with g handles (g ≥ 0), and the nonorientable surfaces Nk, with k crosscaps (k > 0). A graph embedding into a surface means a cellular embedding. For any spanning tree of G, the number of co-tree edges is called the Betti number of G, and is denoted by β(G). A rotation at a vertex v of a graph G is a cyclic order of all edge-ends (or equivalently, half-edges) incident with v.A pure rotation system ρ of a graph G is the collection of rotations at all vertices of G. An embedding of G into an oriented surface S induces a pure rotation system as follows: the rotation at v is the cyclic permutation corresponding to the order in which the edge- ends are traversed in an orientation-preserving tour around v. Conversely, by the GENUS DISTRIBUTIONS OF STAR-LADDERS 3 Heffter-Edmonds principle, every rotation system induces a unique embedding (up to homeomorphism) of G into some orientable surface S. The bijection of this correspondence implies that the total number of orientable embeddings is Y (dv − 1)!; v2V (G) where dv is the degree of vertex v. A general rotation system is a pair (ρ, λ), where ρ is a pure rotation system and λ is a mapping E(G) ! f0; 1g. The edge e is said to be twisted (respectively, untwisted) if λ(e) = 1 (respectively, λ(e) = 0). It is well-known that every oriented embedding of a graph G can be described by a general rotation system (ρ, λ) with λ(e) = 0 for all e 2 E(G). By allowing λ to take non-zero values, we can describe the nonorientable embeddings of G. For any spaning tree T , a T - rotation system (ρ, λ) of G is a general rotation system (ρ, λ) such that λ(e) = 0, for all e 2 E(T ). By the genus polynomial of a graph G, we mean the polynomial 1 X i ΓG(z) = gi(G)z ; i=0 where gi(G) means the number of embeddings of G into the orientable surface Si, for i ≥ 0. 1.3. Overlap matrices. Mohar [17] introduced an invariant that has subse- quently been used numerous times (e.g., [2,3,4]) in the calculation of distri- butions of graph embeddings, including non-orientable embeddings. We use Mo- har's invariant here in our derivation of a formula for the genus distribution of star-ladders. Let T be a spanning tree of a graph G and let (ρ, λ) be a T -rotation system. Let e1; e2; : : : ; eβ(G) be the cotree edges of T , where β(G) is the cycle rank of G. The overlap matrix of (ρ, λ) is the β(G) × β(G) matrix M = [mij] over Z2 such that 8 1; if i = j and e is twisted; > i <>1; if i 6= j and the restriction of the underlying pure m = ij rotation system to the subgraph T + e + e is nonplanar; > i j :>0; otherwise. When the restriction of the underlying pure rotation system to the subgraph T + ei + ej is nonplanar, we say that edges ei and ej overlap. The importance of overlap matrices is indicated by this theorem of Mohar [17]: 4 YICHAO CHEN, JONATHAN L. GROSS, AND TOUFIK MANSOUR Theorem 1.1. Let (ρ, λ) be a general rotation system for a graph. Then the rank of any overlap matrix M for the corresponding embedding equals twice the genus of the embedding surface, if that surface is orientable, and it equals the crosscap number otherwise. The rank is independent of the choice of a spanning tree. For drawing a planar representation of a rotation system on a cubic graph, we adopt the graphic convention introduced by Gustin [13], and used extensively by Ringel and Youngs (see [21]) in their solution to the Heawood map-coloring problem. There are two possible cyclic orderings of each trivalent vertex. Under this convention, we color a vertex black, if the rotation of the edge-ends incident on it is clockwise, and we color it white if the rotation is counterclockwise. We call any drawing of a graph that uses this convention to indicate a rotation system a Gustin representation of that rotation system. The approach here is similar approach to that used for ladders in [5]. In a Gustin representation of a rotation system for a graph, an edge is called matched if it has the same color at both endpoints; otherwise, it is called unmatched. In Figure3, we have indicated our choice of a spanning tree for a generic ladder Ln−1 by thicker lines and a partial choice of rotations at the vertices. b1 b2 bn-2 bn-1 a1 a2 a3 an-1 an Figure 3. A spanning tree and some rotations for the ladder Ln−1. The following proposition facilitates the calculation of an overlap matrix for a ladder graph. The proof is simply to apply the Heffter-Edmonds face-tracing algorithm. Proposition 1.2. In the ladder Ln−1, we choose all of the edges on one side of the ladder plus all of the rungs, except for the two created by doubling, as the edges of a spanning tree. We label the cotree edges a1; : : : ; an, from one end of the ladder to the other, and we label the tree rungs b1; : : : ; bn, from one end of the ladder to the other (as shown in Figure3). Then two cotree edges ai and ai+1, with 1 ≤ i ≤ n − 1, overlap if and only if the rung edge bi is unmatched. GENUS DISTRIBUTIONS OF STAR-LADDERS 5 By Proposition 1.2, the overlap matrix Mn of Ln−1 can be written as 0 x1 y1 1 B y1 x2 y2 0 C B y x y C X;Y B 2 3 3 C Mn = Mn = B . C ; B .. .. .. C B C @ 0 yn−2 xn−1 yn−1 A yn−1 xn n n−1 where X = (x1; x2; : : : ; xn) 2 Z2 and Y = (y1; y2; : : : ; yn−1) 2 Z2 . Our notation X;Y Mn indicates not only that this is a tridiagonal n × n matrix, but also that the diagonal arrays just below and just above the main diagonal are identical.
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